Abstract
We consider the complexity of finding weighted homomorphisms from intersection graphs of curves (string graphs) with n vertices to a fixed graph H. We provide a complete dichotomy for the problem: if H has no two vertices sharing two common neighbors, then the problem can be solved in time \(2^{O(n^{2/3} \log n)}\), otherwise there is no algorithm working in time \(2^{o(n)}\), even in intersection graphs of segments, unless the ETH fails. This generalizes several known results concerning the complexity of computational problems in geometric intersection graphs.
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